When two bodies with curved surfaces are in contact under a force, point or line contact between these bodies changes to area contact, and  3 dimensional stresses are developed  These stresses are contact stresses. A knowledge of contact stresses is important in calculating strength of bearings, gear and worm drives, ball and cylindrical rollers, and cam mechanisms. Typical failures are seen as cracks, pits, or flaking in the surface material.

Contact representations calculated by Hertzian contact calculator are shown below. Circles represent sphere or cylinder depending on contact selection.

Herzian contact (sphere-sphere contact) Herzian contact (sphere-plane contact) Herzian contact (sphere-cup contact)
Sphere-Sphere Sphere-Plane Sphere-Cup
Cylinder-Cylinder Cylinder-Plane Cylinder-Inner Cylinder

The Hertzian contact calculator is designed to calculate contact pressure and contact stress for spherical and cylindrical contact. Von Misses and shear stresses are drawn with respect to the depth from contact surface for Object-1 and Object-2. The formulas used for the calculations are given in the "List of Equations" section.


Parameter Symbol Object-1 Object-2 Unit
Object shape    
Poisson's ratio v1,v2  
Elastic modulus E1,E2
Diameter of object d1,d2
Force F

Parameter Symbol Object-1 Object-2 Unit
Maximum Hertzian contact pressure pmax 1701.5
Max shear stress τmax 508.2 508.2
Max Von Misses stress σmises 1607 1607
Depth of max shear stress z 0.083 0.083
Circular contact area diameter 2a 0.335

Note: Use dot "." as decimal separator.

Object-1 Object-2


Cylindrical contact: The contact of two cylindrical parts where the contact line turns to rectangle area with the presence of a contact force. The area of contact is a narrow rectangle of width 2b and cylinder length l.

Modulus of elasticity (Young’s modulus): The rate of change of unit tensile or compressive stress with respect to unit tensile or compressive strain for the condition of uniaxial stress within the proportional limit. Typical values: Aluminum: 69 GPa, Steel: 200GPa.

Poisson’s ratio: The ratio of lateral unit strain to longitudinal unit strain under the condition of uniform and uniaxial longitudinal stress within the proportional limit.

Proportional Limit: The largest value of stress up to which a linear relation still exist between stress and strain (Hooke’s Law).

Spherical contact: The contact of two spherical parts where contact point turns to a circular area of radius a under the presence of contact force. A contact pressure pmax is occurred on this contact area.

Shear stress: A form of a stress acts parallel to the surface (cross section) which has a cutting nature.

Stress: Average force per unit area which results strain of material.

Von Mises : A theory used to estimate the yielding of ductile materials. The Von Mises criteria states that failure occurs when the energy of distortion reaches the same energy for yield failure in uniaxial tension test.

List of Equations:

List of equations and calculation steps for Hertzian contact stress calculations